机械毕业设计英文外文翻译应力为基础的有限元方法应用于灵活的曲柄滑块机构.docx

机械毕业设计英文外文翻译应力为基础的有限元方法应用于灵活的曲柄滑块机构.docx

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机械毕业设计英文外文翻译应力为基础的有限元方法应用于灵活的曲柄滑块机构

英文原稿

ApplicationofStress-basedFiniteElementMethodtoaFlexibleSliderCrankMechanism

（Y.L.KuoUniversityofTorontoW.L.CleghornUniversityofCanada）

Abstract—Thispaperpresentsanewproceduretoapplythestress-basedfiniteelementmethodonEuler-Bernoullibeams.Anapproximatedbendingstressdistributionisselected,andthentheapproximatedtransversedisplacementisdeterminedbyintegration.Theproposedapproachisappliedtosolveaflexibleslidercrankmechanism.TheformulationisbasedontheEuler-Lagrangeequation,forwhichtheLagrangianincludesthecomponentsrelatedtothekineticenergy,thestrainenergy,andtheworkdonebyaxialloadsinalinkthatundergoeselastictransversedeflection.Abeamelementismodeledbasedonatranslatingandrotatingmotion.Theresultsdemonstratetheerrorcomparisonobtainedfromthestress-anddisplacement-basedfiniteelementmethods.

Keywords:

stress-basedfiniteelementmethod;slider

crankmechanism;Euler-Lagrangeequation.

1.Introduction

Thedisplacement-basedfiniteelementmethodemployscomplementaryenergybyimposingassumeddisplacements.Thismethodmayyieldthediscontinuitiesofstressfieldsontheinter-elementboundarywhileemployinglow-orderelements,andtheboundaryconditionsassociatedwithstresscouldnotbesatisfied.Hence,analternativeapproachwasdevelopedandcalledthestress-basedfiniteelementmethod,whichutilizesassumedstressfunctions.VeubekeandZienkiewicz[1,2]werethefirstresearchersintroducingthestress-basedfiniteelementmethod.Afterthat,themethodwasappliedtoawiderangeofproblemsanditsapplications[3-5]Inaddition,therearevariousbooksprovidingdetailsaboutthemethod[6,7].

Theoperationofhigh-speedmechanismsintroducesvibration,acousticradiation,wearingofjoints,andinaccuratepositioningduetodeflectionsofelasticlinks.Thus,itisnecessarytoperformananalysisofflexibleelasto-dynamicsofthisclassofproblemsratherthantheanalysisofrigidbodydynamics.Flexiblemechanismsarecontinuousdynamicsystemswithaninfinitenumberofdegreesoffreedom,andtheirgoverningequationsofmotionaremodeledbynonlinearpartialdifferentialequations,buttheiranalyticalsolutionsareimpossibletoobtain.Cleghornetal.[8-10]includedtheeffectofaxialloadsontransversevibrationsofaflexiblefour-barmechanism.Also,theyconstructedatranslatingandrotatingbeamelementwithaquinticpolynomial,whichcaneffectivelypredictthetransversevibrationandthebendingstress.

Thispaperpresentsanewapproachfortheimplementationofthestress-basedfiniteelementmethodontheEuler-Bernoullibeams.Thedevelopedapproachfirstselectsanassumedstressfunction.Then,theapproximatedtransversedisplacementfunctionisobtainedbyintegratingtheassumedstressfunction.Thus,thisapproachcansatisfythestressboundaryconditionswithoutimposingaconstraint.Weapplythisapproachtosolveaflexibleslidercrankmechanism.Inordertoshowtheaccuracyenhancementbythisapproach,themechanismisalsosolvedbythedisplace-basedfiniteelementmethod.Theresultsdemonstratetheerrorcomparison.

II.Stress-basedMethodforEuler-BernoulliBeams

ThebendingstressofEuler-Bernoullibeamsisassociatedwiththesecondderivativeofthetransversedisplacement,namelycurvature,whichcanbeapproximatedastheproductofshapefunctionsandnodalvariables:

Where

isarowvectorofshapefunctionsfortheithelement;

isacolumnvectorofnodalcurvatures,yisthelateralpositionwithrespecttotheneutrallineofthebeam,EistheYoung’smodulus,and

isthetransversedisplacement,whichisafunctionofaxialpositionx.

IntegratingEq.

（1）leadstotheexpressionsoftherotationandthetransversedisplacementasRotation：

Transversedisplacement:

Where

and

aretwointegrationconstantsfortheithelement,whichcanbedeterminedbysatisfyingthecompatibility.

SubstitutingEqs.

（2）and（3）into

（1）,thefiniteelementdisplacement,rotationandcurvaturecanbe

expressedas:

wherethesubscripts（C）,（R）and（D）refertocurvature,rotationanddisplacement,respectively.Byapplyingthevariationalprinciple,theelementandglobalequationscanbeobtained[11-13].

Table1:

Comparisonofthedisplacement-andthestress-basedfiniteelementmethodsforan

Euler-Bernoullibeamelement

III.ComparisonsoftheDisplacement-andStress-basedFiniteElementMethods

Themajordisadvantageofthedisplacement-basedfiniteelementmethodisthatthestressfieldsattheinter-elementnodesarediscontinuouswhileemployinglow-degreeshapefunctions.Thisdiscontinuityyieldsoneofthemajorconcernsbehindthediscretizationerrors.Inaddition,itmightuseexcessivenodalvariableswhileformulatingstiffnessmatrices.

Thestress-basedmethodhasseveraladvantagesoverthedisplacement-basedfinitemethod.Firstofall,thestress-basedmethodproducesfewernodalvariables（Table1）.Secondly,whenemployingthestress-basedfinitemethod,theboundaryconditionsofbendingstresscanbesatisfied,andthestressiscontinuousattheinter-elementnodes.Finally,thestressiscalculateddirectlyfromthesolutionoftheglobalsystemequations.However,theonlydisadvantageofthestress-basedfinitemethodisthattheintegrationconstantsaredifferentforeachelement.

IV.GenerationofGoverningEquation

TheslidercrankmechanismshowninFig.1isoperatedwithaprescribedrigidbodymotionofthecrank,andthegoverningequationsarederivedusingafiniteelementformulation.Thederivationprocedureofthefiniteelementequationsinvolves:

（1）derivingthekinematicsofarigidbodyslidercrankmechanism;

（2）constructingatranslatingandrotatingbeamelementbasedontherigidbodymotionofthemechanism;（3）definingasetofglobalvariablestodescribethemotionofaflexibleslidercrankmechanism;（4）assemblingallbeamelements.Finally,theglobalfiniteelementequationscanbeobtained,andthetimeresponseofaflexibleslidercrankmechanismcanbeobtainedbytimeintegration.

A.Elementequationofatranslatingandrotatingbeam

Consideraflexiblebeamelementsubjectedtoprescribedrigidbodytranslationsandrotations.Superimposedontherigidbodytrajectory,afinitenumberofdeflectionvariablesinthelongitudinalandtransversedirectionsisallowed.TheEuler-Lagrangeequationisusedtoderivethegoverningdifferentialequationsforanarbitrarilytranslatingandrotatingflexiblemember.Sinceelasticdeflectionsareconsideredsmall,andthereisafinitenumberofdegreesoffreedom,thegoverningequationsarelinearandareconvenientlywritteninmatrixform.Thederivationoftheelementequationshasbeenpreciselypresentedin[8-10],andthissectionprovidesabriefsummary.

Inviewofhighaxialstiffnessofabeam,itisreasonabletoconsiderthebeamasbeingrigidinitslongitudinaldirection.Hence,thelongitudinaldeflectionisgivenas

whereu1isanodalvariable,whichisconstantwithrespecttothexdirectionshowninFig.2.Thetransversedeflectioncanberepresentedas

Thevelocityofanarbitrarypointonthebeamelementwithatranslatingandrotatingmotionisgivenas

where

istheabsolutevelocityofpointOofthebeamelementshowninFig.2;θ?

istheangularvelocityofthebeamelement;

arethelongitudinalandtransversedisplacementsofanarbitrarypointonthebeamelement,respectively;xisalongitudinalpositiononthebeamelementshowninFig.

2.

Ifweletρbethemassperunitvolumeofelementmaterial;A,theelementcross-sectionalarea,andLtheelementlength,thenthekineticenergyofanelementisexpressedas

TheflexuralstrainenergyofuniformaxiallyrigidelementwiththeYoung’smodulus,E,andsecondmomentofarea,I,isgivenas

Theworkdonebyatensilelongitudinalload,（i）P,inanelementthatundergoesanelastictransversedeflectionisgivenby[14]

Longitudinalloadsinamovingmechanismelementarenotconstant,anddependbothonthepositionintheelementandontime.Withthelongitudinalelasticmotionsneglected,thelongitudinalloadsmaybederivedfromtherigidbodyinertiaforces,andcanbeexpressedas

wherePRisanexternallongitudinalloadactingattherighthandendofanelement,andox

（i）aistheabsoluteeaccelerationofthepointOinthexdirectionshowninFig.2.

TheLagrangiantakestheform

SubstitutingEqs.（5-10）into（12）,andemployingtheEuler-Lagrangeequations,thegoverningequationsofmotionforarotatingandtranslatingelasticbeamcanbeexpressedinthefollowingmatrixform:

where[Me],[Ce]and[Ke]aremass,equivalentdamping,andequivalentstiffnessmatricesofaelement,respectively;{Fe}isaloadvectorofanelement.Whenformulatingthemassmatrixofthecoupler,themassoftheslidershouldbetakenintoaccount.

B.Globalequationsofslidercrankmechanism

Fortheproposedapproachtosolveaflexibleslidercrankmechanism,theglobalvariablesarethecurvaturesonthenodes.Forassemblingallelements,itisnecessarytoconsidertheboundaryconditionsappliedtothemechanism.Sinceaprescribedmotionappliedtothebaseofthecrank,thereisabendingmomentatpointOshowninFig.1,i.e.,thecurvatureatpointOexists.ForpointsAandBshowninFig.1,wepresumethatbothpointsrefertopinjoints.Thus,thebendingmomentsandthecurvaturesatbothpointsarezeros.

SinceEq.（13）isamatrix-formexpressionintermsofthevectorofglobalvariables{φ},theglobalequationscanbeobtainedbydirectlysummingupallofelementequations,whichcanbeexpressedas

where[M],[C],[K]areglobalmass,dampingandstiffnessmatrices,respectively;{F}isagloballoadvector.

V.Numericalsimulationbasedonsteadystate

Therotatingspeed